I have a somewhat embarrassingly big love for mathematics. I find myself talking about it all the time in a variety of social contexts – birthday parties, drinks with my brother’s friends, I seem to take any excuse to talk about the wonders of my field by the horns. I get the feeling that most people don’t understand mathematics as I do (a strangely creative and incredibly powerful area that bridges the gulf between philosophy and science, the study of reason itself for its own sake) and I feel compelled to correct them. Does the subject have an unavoidably narrow appeal? How much is the state of mathematics education to blame? Why do people flinch when I say things like “equation”?

I sat down and read the draft Mathematics syllabus a while ago, but read it again in response to an article I read a few days ago. Though it only addressed the K-10 syllabus, my first stop is the new 11-12 course called Essential Mathematics. Here’s the rationale:

Mathematics is the study of function and pattern in number, geometry and data. It provides both a framework for thinking and a means of communication that is powerful, logical, concise and precise. Essential mathematics focuses on using mathematics to make sense of the world. The emphasis is on providing students with the mathematical skills and understanding to solve problems and undertake investigations in a range of workplace, personal, training and community settings. There is an emphasis on the use and application of information and communication technologies in the course. The course includes investigation of the application of mathematical understanding and skills in workplaces or community settings.

Reading through the meat of the syllabus, you find exactly what you’d expect – measurement, money, pie charts, basic statistics, simple probability. I think this “practical” mathematics is a great course to have available to students, as it fosters the necessary numeracy that is required to navigate today’s complicated world. Innumeracy is just as unacceptable as illiteracy, and is far more prevalent. But I am shocked at the simplicity of the syllabus. How many students are unable to convert between 12-hour and 24-hour time before starting year 11? How many are unfamiliar with the concept of an “average”, or be able to reason through that the chances of getting “two heads” on two fairly-flipped coins is one in four? Are we really making such knowledge optional?

There is almost no content within this syllabus that is not directly addressed in earlier years. It’s as though they’ve stripped things down to “the basics” and want to give failing students another go. The essential problem, however, is that this is not a course that most people would like to teach, and not a subject that most students would want to learn. I’m sure that almost no-one wants to sit in class and learn how to adjust for the GST or draw a stem-and-leaf plot, and I’m just as sure that there are few who would want to be responsible to make these kids do so.

The essential problem, I think, is a misconception of what these basics should be. Students who end up in this course are those who haven’t been engaged with the subject in previous years, and this course isn’t going to change that. It’s like teaching failing kids to read by getting them to read job ads, without telling them about the stories they can read to themselves and others, the world that is opened to them when they can finally understand what those letters actually mean. There is little attempt to make it interesting, as though the needs for such mathematics in society should provide all the motivation students require.

So let’s go back a bit and look at a relatively simple example of something that wouldn’t be taught to a kid doing Essential Mathematics. It’s the Quadratic Formula:

Chances are you had at least one test during school where your ability to use this equation earnt you marks. Now learning to use this equation is hard work; there’s a lot of bits to remember, there’s a “plus or minus” and you have to do that before you divide, and that’s after you’ve done this square root business (surds are tricky). Even then, your calculator might tell you “MA ERROR” when you try to make the equation work, and you don’t know whether you’ve done something wrong or there’s something wrong with the question.

Why on earth do we go to the bother of teaching you how to do this? Almost no-one will ever use the quadratic theorem on even an occasional basis, and those that do already have calculators and/or software that will do it for them. When I teach mathematics to university students enrolled in the bottom two of the five possible levels of mathematics at my university, at least half the students don’t know how to use this equation anyway.

Maybe this like learning to brew your own beer – you don’t need to know how it all works, you can just buy it from the store, but it’s good to know “just in case” and those that get good at it can make lots of money. Hell, it can be a fun hobby and maybe you can earn some money, but if you don’t get it, then it’s not really that big a deal. This is certainly the line most mathematics teachers I’ve come into contact with use when defending their subject. The students are told that you’ll get a better job if you know it, somehow, probably. Also, it scales really well in the HSC, so you should do maths.

But I think that misses the mark. The obvious parry heard from many students is that while it may be useful for some people, it’s not going to be useful to them. They’re going to be an artist, a novelist, a landscape gardener, work in a call centre, when are they going to use this formula? And unfortunately for the state of mathematical education today, I don’t blame them for thinking this way. The mathematics syllabus is becoming more and more skills-based, meaning more and more disconnected pieces of highly specialised mathematical knowledge most useful for the job market. This means lots of calculator use at lower levels, and only statistics and hardcore calculus (both very good for making money) for those who are more interested in the subject.

Above all, the emphasis is on usage. This objection can be applied equally to many other disciplines – when does knowledge of Shakespeare, or the Napoleonic Wars, or the fact that we’re made of carbon affect your daily life? When are you going to use that knowledge? But as far as I can tell from my own personal experience, this objection is not as prevalent elsewhere. It’s as though the importance of learning the subject for its own sake is so embedded in the way the subject is taught that you don’t even need to ask, that the subject is engaging and interesting enough that the students don’t want to ask.

An additional and related complication is quite obvious to me in the way these problems are often phrased:

Boooooorriiiiiinggg…….

Not only is this boring, but it’s also unrealistic. Who decides what area their path should have? Whilst being able to take words and turn them into mathematical formulae and then solving is an incredibly important skill, I can see why this kind of question really turns students off. They’re technically presented with exactly the amount of information they need to solve the problem, and no more, though situations are rarely like this. Instead of engaging their problem-solving skills in an interesting way, showing how mathematics can make sense of complex situations, the single skill of “turn words into formula” is tested in a simplistic and frankly dull way.

Mathematicians try and demonstrate how their knowledge can be used in the real world, but all it serves to do is to show how out of touch they are. It fosters a belief that the mathematical world is a separate, Platonic world, where there is RIGHT and WRONG and nothing in between, ugly and inhuman. Who thinks like this? Who asks these stupid questions?

We need to start thinking differently about what mathematics can and should inspire in people. Imagine you’ve invited your friends over and cracked open a brand new board game.

So good, this game.

You grab the rulebook, and start reading. After you’re done, you’ll hand it to someone else, until everyone’s up to speed. You pick your colours and away you go.

Even though you all start from the same ground state (you all know the rules and haven’t read any strategy guides), one of you will win. Even if there’s an element of luck, good players will win more often on subsequent playthroughs. There is, somehow, not enough information – and yet you’re expected to do the best with what you’ve got. Players who play enough may start to create “house rules”, alterations of the foundation to maximise play experience. What is going on here? What makes one foundation better than another? What makes one person better at the game than another?

There’s an analogous skill, one that is equally important – what happens if you’re presented with lots of information (in the form of conflicting strategy guides and advice from experience players), and you have to decide what’s relevant? How do you decide what makes an effective model for victory?

Whatever the answers to these questions are, they are what mathematicians like myself want to find, and a desire for such answers is what we want to inspire in others. The applications are instantly infinite – we almost always have too much or too little information, and we have very little sense of what’s going on “under the hood” and what that means for the way things play out. The aim of this approach is to help people correctly identify the foundational assumptions (“the rules”) of a situation, asses the possibilities, and make the most of it. Everybody does this every day, to varying degrees of effectiveness. The skills of argument, of comparison and contrast, of quantifying and extrapolating, pattern recognition and modelling; in short, the ability to make distinctions using reason are so important and universal, and mathematics is poised and ready to help people learn how to do this better.

Now other subjects teach this love of logic too. I loved history for the same reasons I loved mathematics – you tried to get a sense of the past, of people’s motivations and actions from the texts and facts available, where weighing up evidence (establishing the foundation) was intimately linked with constructing the best argument (the finished product). What drew me away from history and humanities in general was that I couldn’t stand the cheating, so to speak – the conflation between medium and message, where gifted speakers or writers could have undue influence on others through careful window-dressing of bad arguments. As a shamefully gifted cheater, I found my basic essay skills plus a light drizzling of facts could pass me through a book review with a minimum of time spent actually reading the book.

Some people have an “ear” for languages, and some people can barely speak their own, and the analogy holds just as well for mathematics. Mathematics requires a lot of discipline, and that doesn’t help its attractiveness, but it really is like learning a language. Add to that the fact that the “rules” to a lot of things we grapple with nowadays are very complicated and not very intuitive – most of the time, they’re counter-intuitive. Here’s two nice examples where we go nowhere near the world of physics.

Suppose you’re on a game show and you’re given the choice of three doors – behind one is a car, and behind the other two there is nothing. The car was placed randomly behind the winning door before the show. You pick your door, but the host doesn’t open it yet – instead, he opens one of the doors you didn’t choose with nothing behind it. Now the car is either behind the door you originally chose, or the other unopened door. You’re given the chance to switch your preference. Should you do so?

This is the famous Monty Hall problem. Identifying what is important in this problem is hard, and the answer is even more surprising – switching doors doubles your chances of success!

Suppose you’re in a room with a group of people, and you each call out your birthday in turn. How many people would you need to get into a room to make it 50% likely that at least two people share a birthday?

This is the equally famous Birthday problem, and the answer (23) is quite surprising – especially considering you need 367 people to make it a sure thing!

Even though you don’t need words for numbers to have an innate sense of number, the level of abstraction required to do this kind of thinking is quite intimidating, and humans are not custom-built to do this kind of manipulation. You have to do it every day to make your brain think in these new, unnatural ways, and nothing is going to make it easier. Mental discipline is a good thing to encourage, it must be said, but sometimes students only see the stick and don’t get to see the carrots that keep people like me going.

For example, I was having a discussion with a (non-maths) friend of mine about how cool maths is while we were out for dinner (any excuse will do, remember?). He recalled the story of Isaac Newton, who invented Calculus. He just made it up. At the same time as Leibniz, sure, but he invented it. How cool is that? It’s one of the most powerful analytical tools today, and it wasn’t around before he was. Why did he do it? What possessed him to make it the way it is – could he have made it a different way? Why is the way it is today “the best” way? What do we even mean by that?

How about the statement “This statement is a lie”? What is going on here – linguistic trickery, or something more fundamental and interesting?

How about fractals – Benoit Mandelbrot asked the seemingly simple question “How long is the coastline of Britain?” and came up with the answer “Infinitely long“, and he could say it with a straight face. Paradoxes and puzzles are compelling, and fractals have the bonus features of being really pretty and engage students with technology!

Watch this in HD with the lights off!

Maybe you knew something of these wonderful ideas, and maybe you learnt about them in high school, but I doubt it. I was blown away by this stuff when I got to university, and I can’t understand why students don’t learn about this. This kind of backwards thinking is so stunning, and appears to be unique to mathematics. They don’t teach you the cool stuff, or even give you a hint that it even exists. Teachers battle to tell students how their learning will earn them money or get them better marks, and that’s the only way the conversation is developing – and the syllabus along with it. I’m going to quote from the article I linked before. I agree with it in its entirety.

“Just as children best learn to read by experiencing the joy of great stories, they best learn mathematics by experiencing its beauty and the joy of mathematical play. But in this curriculum there is little sense of the fun and the beauty of mathematics. Not a hint of infinity, of the fourth dimension, of Moebius bands, of puzzles or paradoxes.

Why? If mathematics can be taught as ideas, as something beautiful and fun, then why is it not being proposed? Because it is difficult to do. To teach real mathematics makes demands on the teacher, and it is risky.

What is proposed is little more than a cowardly version of current curriculums, a codification of the boring, pointless approach – which is “safe” but which has already failed a generation of students.

The draft curriculum begins by declaiming the beauty and intrinsic value of mathematics, and the elegance and power of mathematical reasoning. But as a means of unfolding all this before our students, the proposed curriculum is a feeble tool indeed.”

If you look at what engaged the founders of mathematics (Euclid and his geometry, Newton and physics, Gauss and number theory), you’ll find the means to encourage students to participate. You could talk bright students through the Millenium prizes, to show them what captivates mathematicians to this day – even get them to look at Hilbert’s problems, which guided the course of mathematics throughout the twentieth century. For other students, teaching them about puzzles and games and paradoxes would be easy, fun and valuable. Instead, we’re teaching them how to use a calculator and calling it a day. Trying to engage students by focussing on how society sees and uses mathematics is going to force this negative feedback of disengagement to continue.

There is such a shortage of mathematics teachers in Australia that you can become a high school mathematics teacher with six months of training, having only completed 2 unit mathematics yourself (source). You will be called on to teach classes that you yourself only need to have passed, and there is no requirement for any higher training whatsoever.

I say again, you can become a high school teacher, having only passed 2-unit. Who thinks this will solve anything? If you’re running out of mathematics teachers, the solution cannot be to scrape the bottom of the barrel, so to speak. Something more drastic needs to happen.

Listening to trainee teachers talk about mathematics in the classroom was one of the main reasons I dropped out of my Masters of Teaching and went into tertiary study instead. There was such a dearth of passion for the subject. Most students were surprised that I had done advanced level mathematics, saying they found that stuff “too boring”. I cannot understand how making it easier for poor quality maths teachers to enter the system is going to solve anything, and with the syllabus in its present state, I can only see this trend continuing.

Mathematics is beautiful, fun, compelling, dramatic, creative and human. Mathematics education is crude, oriented towards the job market and painfully dull. And things are only going to get worse.

Well, not if I can help it. I’m going to spend my life trying to fix the perception of mathematics at every level of society. It is my mission, my calling, and if I can make it work, my career.

Recently, I went to the 10th Representation Theory of Algebraic Groups and Quantum Groups conference, held in Nagoya, Japan. Afterwards, I travelled to Tokyo to meet a friend of mine, who gave me a massive bonus to my Adventuring (Tourist) roll. Here follows my tale of of culture shock (and awe), detailing my delightful discoveries in the strangest land I have ever travelled.

First of all, the mathematics conference was brilliant. So many amazingly smart people, chalking and talking their way to my heart:

The critical level representation – efforts in the arena of the Geometric Langlands Program

It was a great way to find out what was important to other people. As a student, I want to know what others find interesting; that way, my own research efforts might stay relevant to others. When I find someone I want to work with in the future (read: apply for a job), I want to have something substantial to offer my collaborators!

The conference was entirely in English, and the atmosphere was warm and welcoming. Also, the air conditioning and free wireless were pretty great! At the conclusion of the conference, the organising professors sang Karaoke for us, which was a pleasant surprise – though slightly alarming. Not to be outdone, the younger attendees all went out to an 8-story karaoke complex, paid ~$23AUD for two hours of singing and all-you-can-drink fun. I sang some Silverchair, the smartest guy in the room (not me, btw) mangled Amy Winehouse so badly that I cried with laughter, and the Italians gave a great rendition of Killing Me Softly. We got very drunk, and learnt about how Japanese can make a pretty decent beer. Not wine, though, omg.

A few of us kicked on to a lovely sake bar, where you could get awesome drinks and a little bit of food, all delivered to your table. I really liked what little of the “drinking house” culture in Japan I saw in Nagoya.

Before that, I was mostly on my own, in a strange city where nobody could understand a single word I was saying. I was determined to not eat fast food, so I tried as many restaurants as I could. The Japanese accent is everywhere in their cuisine, and emphasis on non-Japanese authenticity is low – the “Authentic Southern Fired Chicken” I ordered from a fully-decked-out-even-with-cowboy-costumes-and-Cheers-playing-on-the-TV American-themed bar was regular Chicken Karaage with chilli powder. Most of my meals were achieved through pointing and hand gestures, but delicious, cheap food is abundant. I never spent more than ~$9 on any meal, and it was all extremely filling!

But the Internet was very difficult to find. There was a coin-operated internet machine in the hotel lounge:

Usage is easy! If the coin of 100yen is thrown in, it will be OK!!

…but it was terribly old and slow. I decided to go in search of better internet, and experienced racism directed towards me for the first time in my life. I followed signs (in English!) to Internet cafés, and was told on two separate occasions that their establishment was “No gaijin, Japanese only”. The third cafe I went to was reached by lift, and the lift doors opened to reveal… another locked door with an intercom, right up against the lift. I pressed the button and tried to gain access, but lost my nerve when the doors started closing and I still hadn’t heard any English. Eventually I found a place that would take me, but it was just so surreal.

Like this sign:

I understand the first four. But the fifth is… do you sit here if you’re crying?

People were somewhat reluctant to sit next to me on trains Nagoya, and not speaking the language was so, so hard. It was very isolating.

Oh, back to the technology – I think I was expecting Japan to be like Australia, only at a higher technology level; what I found was a higher prevalence of somewhat gimmicky technology:

Happy poopy time

…but overall, technology didn’t seem substantively more advanced. The trains were great, though, once you puzzled out where you were and where you wanted to be and what platform you had to be on to get the right coloured train – they came about once every five minutes, and moved a lot of people.

I also saw a castle that was being rebuilt, having been destroyed in WWI. It was pretty impressive:

There were many (~6?) of these. It’s as steep as it looks!

….and the moat was cool:

Sure, no water, but try getting through that without getting shot at first!

And so my time in Nagoya drew to a close, as does this post. I’ll continue my Japan adventures in the coming weeks, still to come:

A dance club catering to the geeks of Japan – they use .gifs to enhance their music, for real.

Engrish, oh lol

Bullet trains!

More of my clashes with the oddities of Japanese culture -“They don’t have street signs? What do you mean, they don’t have street signs?”

The Four Colour Theorem is a famous problem in mathematics. It is very simple to express, but proved very difficult to solve. The method of solution was both new and controversial, to the point where some mathematicians still consider the question unanswered.

Early Formulation

In 1852, a graduate student by the name of Francis Guthrie noticed something interesting when he was colouring in a map of the English counties. He discovered that he could colour it all with only four colours, so that no two regions which were adjacent (that is, shared a common border) had the same colour:

As any good mathematician would do in this situation, he generalised the question – is this true of all maps? If some catastrophic occurrence befell the English counties and all the lines were redrawn, would he still only need four colours? What about a completely abstract partitioning of the plane, such as this:

Yep, still only four colours!

He simplified things down a little bit – countries that had more than one connected piece were disallowed, to avoid situations like these:

There is no way to colour this map with four colours, if both the regions marked A are to be the same colour.

It’s easy to show that at least four colours are needed (try it yourself!), but he couldn’t answer why only four were needed. No doubt he tried many different combinations, scribbling in notebooks and trying to find an easy counterexample. Instead, he passed the problem on to someone else.

Difficulties

Augustus De Morgan was a rather famous mathematician – anyone who has studied complex numbers will no doubt remember De Morgan’s Laws, not to mention his work on solidifying the method of proving by induction. He was able to prove a simpler result, which he hoped to generalise: any map with five (or fewer) regions could be coloured with only four colours. However, he couldn’t prove it for an arbitrary number of regions.

The problem was captivating enough, so it was passed around the mathematical community, attracting attention from brilliant minds such as Sir William Hamilton and Arthur Cayley.

After the problem was published in the journal Proceedings of the London Mathematical Society, a man by the name of Arthur Kempe struck upon a brilliant idea.

First he proved that any map has a region with five or fewer neighbours (which is quite remarkable, considering the generalities in which we’re talking). He described a process of shrinking which he claimed was reversible: if a region has three or fewer neighbours, we can shrink it away so that the map now has one less region to colour. We shrink everything down until our map only has four regions, colour those arbitrarily, and reverse the process:

The grey regions are shrunk one at a time until we have only four regions – they are coloured, and the shrinking process is reversed.

Everyone was happy for 11 years, until Percy (!) Heawood found that the process wasn’t as reversible as everyone had assumed: a region with five neighbours couldn’t be shrunk like this (so that you could colour the rest and put it back nicely), and that was an unavoidable for some maps. The proof was sunk.

And yet, no-one could find any map that required more than four colours. They tried everything they could think of, but mathematicians are always (or should be) mindful that they are rather meagre minds in the light of the massive order and reason they are wrangling with – perhaps people just weren’t clever enough to come up with the required counterexample!

The language of graph theory was being developed, and was nicely adapted to this problem. Instead of considering countries and regions, the graph theorists considered coloured points, with lines between them representing adjacency:

With the results of graph theory now at their disposal, mathematicians quickly dispatched the Five Colour Theorem – I read the proof, and actually reproduced it rather faithfully in my exam, when I took Graph Theory in second year. However, the Five Colour Theorem wasn’t much help – moving from a graph that is five-coloured to one that is four-coloured often requires a lot of recolouring, as can be seen in this example:

A five coloured map.

The same map, now four coloured – note that the central blue region is now green, and its lower left neighbour is now blue.

In 1922 a man by the name of Philip Franklin proved that any map with 26 or fewer regions could be four-coloured. His proof used an idea called a reducible configuration, first introduced in Kempe’s proof (though not named as such). The idea was similar – take a number of adjacent regions, remove them, and four-colour the remainder. If you can put the regions back, regardless of how you four-coloured the remainder, then the regions are said to form a reducible configuration. Note that this is a generalisation – we are removing groups of regions, rather than one-at-a-time like Kempe proposed.

The idea was therefore formulated that would later be the starting point of the proof: prove that every possible map has a reducible configuration. Then, when you remove it from a particular map, the new (and simpler) map will also have a reducible configuration, and you can repeat the process and we have our result.

But what do these reducible configurations look like? It was demonstrated that a region that has four or fewer neighbours was reducible, but was that it? Could this turn into a proof?

A new age of proof

In 1970, Wolfgang Haken began his search for the full list of reducible configurations. Most estimates put the number at about 10,000, but the number might have been anything up to infinite (therefore frustrating his search for an exhaustive list). Working with another mathematician, Kenneth Appel, they developed computer software to help them find this list. They wrote their program, fired it up, and hoped that it would one day terminate and say “Here it is, the full list!”.

In just under 1,000 hours (about 6 weeks), the process stopped at 1,936 reducible configurations. They had found the list. Every map had at least one of these configurations – if you removed it to make a simpler map, the new map would contain one of these 1,936 configurations which you could take out. Eventually you could get down to just four regions, colour those and reverse the process. Thus the theorem was proved.

It was one of the first computer-assisted proofs to be presented, and as such gained a lot of attention due to the problem’s age – computers had achieved something humans were unable tackle with over a hundred years of pondering.

Scepticism and acceptance

Modern computers can do Haken and Appel’s computation in just under an hour, but the computation is so hard that no human could ever hope to do it, even if they devoted their entire life to this theorem. Mathematicians were very wary of this result – it comes down to trusting that the computer has done things the way you wanted it to. Computers are rather predictable machines, so most mathematicians do accept the proof today, but there are still some that remain sceptical.

Perhaps a bigger problem is that the proof doesn’t give us a very good understanding of what’s going on – what is it about a plane that it can be coloured in this way? So far, the only answer we have is that we’ve tested all the possible cases. Mathematicians want to be able to understand things like this, to assist in our generalisations.

If we are colouring a sphere or a cylinder, the result is the same. But what if we were to colour a torus, or donut shape? In that case, we actually need seven colours:

Seven regions, each with six neighbours, if we were to roll it together like so:

…so we know we need at least seven colours. Is seven all we need? Again, we’d need to feed all of this into a computer, which would probably give us an answer. The method of proof is therefore “give the computer a surface, and it will test it for you”, rather than a nice formula to describe how many colours you need, no matter how your object is sitting in space. What about a donut with two holes? What about a Mobius strip? The answer is the same – ask the computer.

What about higher dimensions? Again, ask a computer. We don’t have a good knowledge of what the connection is between the shape of an object and how we colour it – we just know that if we do a whole lot of calculations, a computer will give us an answer.

The fact that four colours is the maximum number of colours we need is now clear. However, the way that the fact was proven makes it almost useless to help us in our understanding of what’s really going on. The problem is therefore still being worked on to this day, so that one day humanity may be able to understand without the aid of a computer. We’re working towards a better answer, even though we already know what it is.

(I predict that computer assisted proofs play an increasingly prominent role in modern mathematics. As computer power begins to outstrip the capacity of the human brain, we will want to answer questions beyond our own power to solve. We won’t be able to help ourselves!)

Mathematics is often portrayed as a black-and-white, right-or-wrong subject. Whilst this may be true for much of high school mathematics, it is clearly not when it comes to problems like these. What it means to be right, how you’re right, how you go about finding answers and whether you can use your methods to answer bigger questions is far more important than simply knowing whether your attempts agree with the proverbial answers in the back of your textbook!

For the past few weeks, my life has been devoted to the teaching of mathematics. I’ve been lecturing Calculus at the Summer School, and teaching school leavers 2 unit mathematics in the form of an intensive bridging course. I’ve had a chance to interact directly with over 100 students, and their attitudes towards mathematics were similar to those I’ve encountered while tutoring privately. I thought I’d blog about the difficulties many students face, and how I’ve attempted to tackle them.

It’s counterintuitive and (therefore) hard

Mathematics attempts to deal with some rather abstract concepts. Most people will know a bit of mental arithmetic, but even something as simple as the “times tables” are drilled into us at a young age as part of their mathematical training. People tend to remember some very well, but not others – the 6’s, 7’s and 8’s are particularly confusing for most, since their ‘pattern’ is not as easy to identify (even though the pattern they make is rarely emphasised – rote learning is the key, for some reason).

Asking students to add fractions is an order of magnitude more difficult – it makes sense (according to how they think about addition) that 1/2 + 1/3 = 2/5, rather than the correct answer of 5/6. The necessity of generating a ‘common denominator’ is not something students understand. The fact that 2/5 is less than 1/2, and that adding a third to a half shouldn’t make it less than what you started with, or what 2/5 represents, doesn’t occur to them. What’s worst is when they don’t even know whether they’re wrong or right. They just write stuff down and hope it turns out to be correct, using rules that are often completely inappropriate for the given situation. This stage, which I encounter frequently, is a classic case of a mathematical collapse. The student is left feeling helpless and stupid, and the more you feel like that when you’re engaged in any activity, the more likely you will be to avoid it in the future.

And that’s just fractions, something you should have mastered (according to the syllabus) by the end of year 7. Once we start talking about the trigonometric functions like sine and cosine, students can’t put all the pieces together. It’s a function, rather than a number – something students cannot have automatic intuition for. Functions are mathematical machines, invented in our brains. They don’t exist anywhere else but on our calculators (who have a computer-designed approximation anyway). Thus you cannot add sine or divide by sine without knowing what you’re doing – applying the rules designed for numbers aren’t going to work here.

What is sine? It’s a function of angle that gives the height of a triangle inscribed on a unit (radius 1) circle with the given angle. Here’s a nice .gif:
What sine does is not so hard to explain, and cosine is nicely symmetric – it gives the horizontal distance rather than the height of the same triangle. It’s apparently hard enough, though, that most teachers will resort to “It’s a button on your calculator that you push. It’s equal to Opposite over Hypotenuse”. People who get this explanation are receiving explanations on the order of “World War 2 happened because Nazis were evil”. Easy, but oh so wrong. Perhaps you are dumb enough to need that kind of oversimplified explanation, but I sincerely doubt it.

The tools that mathematicians have developed to understand things such as massive quantities, probability, algebra and calculus are deeply unintuitive. Every method is developed with particular objects in mind, and generalising to other objects is hard – if it works for one thing, why doesn’t it work all the time? These questions are hard to answer, and even harder to answer in a 40 minute period with 30 other children in the class. This leads me nicely into:

Mathematics relies on generalisations of previous knowledge

Whenever I start tutoring students, whether they’re in year 7 or year 12, I always start with fractions. I’ve never had a student, not once, who could perform all the (rather elementary) operations without any mistakes. When you ask students to deal with something like fractional indices, you’re assuming that they can work with fractions. How can a teacher even begin to talk about fractional indices while their students can’t add a half and a third?

Many students get a blank look on their face when they find out they’re wrong after attempting to apply their reason to a problem. They have no idea why they’re wrong, but they don’t know what they don’t know. They gaze off in a helpless stupor as you try to explain to them what they should have done. They’re sure it all makes sense, but they no longer have any idea of why one approach will give the answer, whilst another will not.

Mathematics relies on the foundations that were (or should have been) laid in previous years. If you miss out on those lessons in fractions, or spend an entire period thinking “I hate my teacher” rather than concentrating on the concept of negative numbers, you start to fall behind. Most students never recover to a point where they can enjoy maths again, because it will forever be an uphill battle.

It’s my challenge as a teacher to locate these misunderstandings, correct them and press on with the more difficult material before the stupor sets in.

Mathematics is poorly motivated

“When are we ever going to use this?” is a question I get asked all the time. Mathematics is the kind of subject where the motives aren’t always clear, but the question always seems like an excuse for not working or succeeding whenever my students ask it. Clearly, if it’s all just fanciful nothingness, it’s not so bad if I can’t do it, right?

Sure, students might not be able to see exactly how to use mathematics to improve their life, but I don’t see how it’s any different to how other subjects work. Any benefit you will get out of learning most subjects will be tangential. You never hear:

“This perfectly executed sentence got me my job. Thank you English!”
“Thank god I knew about the way castles were fortified against sieges – the zombies would have eaten me otherwise! Thank you History!”
…and so forth. In reality, it’s more:
“I’m a more critical thinker now, and have good analytical techniques when it comes to messages being communicated to me. Thank you, English/History!”
…which is not something you can quantify.

Mathematics is sometimes represented as learning to be like a calculator or a computer. Whilst mental computation is a useful skill, it’s certainly not what the subject is from about year 7 when we start talking pronumerals and make claims about all numbers. I think this is an attempt to quantify the skills you are learning, rather than appreciating the abstract and tangential benefits you will learn to master.

The response in recent years seems to have been to make mathematics very connected to the real world. Taking a look at the General Mathematics syllabus, you can see their attempts – there is a large portion devoted to financial mathematics, the most boring kind of maths. Seriously, depreciation models. Any interesting subjects (such as spherical geometry, probability and the normal distribution) are dumbed down to the point where virtually no mathematical understanding is required – simply some real-world flavour coating on easy substitution (put x into the equation in the formula sheet) problems. I can’t tell you how frustrated it makes me when I flick through the textbooks of my students and feel so uninspired. If I can’t feel excited about this stuff, how can they?

I think there’s a problem at the heart of mathematics that makes it all seem a little pointless, and it goes deeper than just the “when will I use it in the supermarket” question. Mathematics is all about deductive logic – making conclusions from a given set of premises:

In a sense, the conclusion was already hidden among the premises, that is, in order to communicate the three statements above, I only need to communicate the first two. A mathematical problem is therefore about discovering true things (that are very often profound) by looking at what premises and conclusions you have already reached or discovered, and about the economy of expression.

This is very different to inductive logic, which is what science is. It attempts to take what you know and generalise outwards from there, for example:

Premise: All the whimples I’ve ever met were awesome.
Conclusion: All whimples are awesome.

I find deductive reasoning all the more fascinating and wonderful, which is why I chose to become a mathematician (even though science is WAY AWESOME). However, I can see why many students might feel a sense of “why bother?” when it comes to mathematics. There is only one answer, it’s there, and someone else can figure it out, really. Or you can just copy it from Bob – the Back of the book – and it’s not really that big a deal. The answers have no real meaning in and of themselves, so what’s the harm in copying?

The great profundity that I find in mathematics is its ability to say such powerful things about such a general class of objects. By starting with a few premises, so much can be artfully deduced. Mathematics truly is an art of elegance, much like poetry – it’s all in the selection and expression of thought. How we learn mathematics is also similar to the way you learn to play a musical instrument, or even learn to dance – drills and scales and boring repetition are all designed to get you used to the tool (the instrument or your body) that you’ll be using for profound expression later on.

… but it’s hard to tell that to the girl who’s got to pass MATH1011 so she can go be a vet and never think about x’s and y’s ever, ever again. Whilst I think that mathematics is one of the most essential things in life, and that it teaches you about proof, how to be sure, how to listen and how to deal with so many problems you might face in life (be they logical conundrums or even moral quandaries), deals with the secrets of the universe and holds the most exquisite beauty possibly conceivable by the human mind… I can honestly (and grudgingly) say that you could do without formal training. In the same way that some people just don’t like music or reading or television or the internet and instead find something else to fill their time, I can see why you might just not enjoy anything about mathematics. I am confused by this response, but I can accept it. Give me 10 minutes, though, and I think I could tell you something mathematical that you’ll find quite awesome (I hope to do this for my next blog post!).

Mathematics is Inhuman

Mathematics is all about discipline. It teaches you to ignore your intuition on many levels, instead forcing you to use your reason in very specific ways in very specific contexts. It teaches you to keep all the facts and assumptions in mind when you’re talking about anything, as the most basic properties may hold the key to your success. It really does stretch the brain out to places humans really haven’t needed to go (in terms of evolution), so the fact that we can do it is truly amazing.

However, it is inhuman in the sense that there is never any grey areas. Your answer is either right or wrong – at least in all mathematics before third year pure. You cannot argue your way out of mathematics or ‘fudge’ it with natural talent the same way you can with many of the humanities – remember those times you didn’t read the book but still got a good mark on your book review? Or wrote an essay that was kinda crap but you made it sound impressive with your words? Not so in mathematics. If you don’t know SOHCAHTOA, and what it means, then there is no way you can get any marks in a trigenometry test. It’s not so much about the inherent difficulty of the subject, but more about what questions are asked. I try and encourage in my students an open, inquiring mind that forms its own mental models of what’s going on, their own mental methods for solving problems. I try and ask questions that have multiple answers, using open questions rather than “what is s when t=1”. By encouraging them to actively engage with it in a more organic way, they need less help in staying motivated.

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Those are some preliminary thoughts. I’ll work more on this stuff for more postings in the future, perhaps when I’m not feeling so sick and rundown. I sure do want to talk about general numeracy as well as the wonders of mathematics a whole bunch. Please comment and let me know of your own personal feelings about and encounters with mathematics, because I’m always, always interested!

In the meantime, I’d like to keep you abreast of the awesome internet phenomena, such as kittens inspired by kittens:
….that video has exploded across the internet in just a few days. Funny how a video like that can lie dormant for months, and then the right connections are made and the right sites hear about it and before you know it your video has over 2 million views.

Also, sorry about the late post. Next Sunday, we’ll have Finn delivering a post, and Aidan the Sunday after that.

I really, really wanted to talk about maths today. I’ve been teaching it for (I kid you not) about 14 hours every weekday for the last few weeks, and it’s been pretty intense.

I feel like I’m starting to get a feeling for what most people find difficult, frustrating and boring about mathematics. I think I’ve also got some pretty awesome insight into the world of maths, how people learn it and how it should be taught.

However, I have 78 (!) new non-spam emails in my inbox and a class to teach very soon. When I’m done, I am going to go home and catch up on my sleep and try not to have a brain haemorrhage. I’ll save my insighty goodness for another day, and instead post some awesome mathematical images that I’ve found for your amusement and delight.

The first is a mathematical diagram representing what is called the projective space PG(3,2), and I think it has a pleasing fiveness to it:

I got this from a paper called “Pretty pictures of Geometries” by B. Polster. He says:

“Why are good pictures important? Two of the main reasons that come to mind are the following:

-To convey some of the abstract beauty of the objects we study to people outside our field. This seems to be especially important today as it becomes more and more important to “justify” and “sell” the kind of research we are fascinated by.

-Many of us think in terms of pictures of various degrees of abstraction. The kind of pictures we want to concentrate on in this note are immediately accessible and can serve to lure students into studying incidence geometry and as a first step in teaching students pictorial thinking in geometry.”

The kinds of mathematics I’m engaged in deals with abstraction as well. I study Algebra, not Geometry, so the beauty I find there is almost always unable to be drawn, though there are a few interesting diagrams. Here is a picture representing the “root system” of the algebra G_2, the object I studied for my Honours thesis:

What it ‘really means’ is rather complex, but for the moment, it’s enough to say “oooh, pleasing sixness! Hey, isn’t that the Star of David?” or something like that.

(I don’t know what this one is called, but I like how organic it looks)

Each of these fractals have a property called “Self-similarity”, which means that you can find copies of the whole or parts of the image, in miniature, throughout the fractal. If you look closely, you’ll see the self-similarity in each of those fractals above. You can even download and try this absolutely fantastic fractal zoomer called XaoS, which lets you both zoom in on parts of a fractal to see this self-similarity in action, and form Julia sets like this one:

Here’s how to represent a 4-dimensional object using a 3D image, utilising time as a replacement for a fourth spacial dimension:

I find this rather hypnotic! I’ve heard that those who study geometry get a ‘feeling’ for how things look in 4D space by doing things like this in their brain for all kinds of 4D shapes. Wat.

I have to go to class now, so I’d better cut this short. I’d like to leave off with a question, though. Consider this image:

Do you see her rotating clockwise or anti-clockwise? It’s important to note that nothing is rotating – it’s just a moving 2D silhouette. Our brains interpret the image as rotating, but you can get her to reverse direction if you concentrate hard enough. You have to force your brain to interpret the data it’s getting in a completely different way, and it helps to figure out what parts of the image are making your brain say “this is a rotating woman!”.

Tell me what you see, and if you can reverse her rotation in your brain! Next week, I’ll talk more about maths.